In three dimensional Cartesian coordinates the Hamilton operator, del, is written as

$\nabla= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix}$

The divergence of a vector field $A$ is written as

$\nabla \cdot A= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \cdot \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} =\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z} $

The curl is writtten as

$\nabla \times A= \begin{pmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{pmatrix} \times \begin{pmatrix} A_x \\ A_y \\ A_z \end{pmatrix} = \begin{pmatrix} \frac{\partial A_z}{\partial y} -\frac{\partial A_y}{\partial z} \\ \frac{\partial A_x}{\partial z} -\frac{\partial A_z}{\partial x} \\ \frac{\partial A_y}{\partial x} -\frac{\partial A_x}{\partial y} \end{pmatrix} $

I am trying to write these three equations for generalized coordinates using the Einstein summation convention and the basis vectors $e_1$, $e_2$, and $e_3$ for the scalars $x^1$, $x^2$, and $x^3$

So far I have the equations for gradient and divergence as

$\nabla= e_\mu \frac{\partial}{\partial x^\mu}$

$\nabla \cdot A= e_\mu \cdot\frac{\partial A_\mu}{\partial x^\mu}$

Are these equations correct? What would the equation for curl be?

  • $\begingroup$ @DomDoe that is the Hamiltonian operator $\endgroup$ – Ryan Parikh Oct 29 '19 at 8:48
  • 2
    $\begingroup$ No, they're not correct at all (and the second one doesn't make sense to begin with). There's a Wikipedia page describing the formulas for typical specific cases and one with the general case. Why don't you start there, and come back here if you have specific questions? $\endgroup$ – E.P. Oct 29 '19 at 8:53
  • $\begingroup$ The specific pages for grad (en.wikipedia.org/wiki/Gradient), div (en.wikipedia.org/wiki/Divergence), curl (en.wikipedia.org/wiki/Curl_(mathematics)) also have generalized versions. You will need some grasp of what vectors and co-vectors are, what is metric, what is covariant derivative, what is the connection, Levi-Civita relative tensors, and generalized Kroenecker deltas. This is for completely arbitrary coordinates. Simpler expressions are available if you stick with curvilinear coordinates $\endgroup$ – Cryo Oct 29 '19 at 11:37
  • $\begingroup$ @Cryo "You will need some grasp of [long list of things that are arcane and mysterious right up until they are easy and obvious]" OK. That made me chuckle. $\endgroup$ – dmckee --- ex-moderator kitten Oct 29 '19 at 15:08

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